We present a framework to describe completely general first-order perturbations of static, spatially compact, and locally rotationally symmetric class II spacetimes within the theory of general relativity. The perturbation variables are by construction covariant and identification gauge invariant and encompass the geometry and the thermodynamics of the fluid sources. The new equations are then applied to the study of isotropic, adiabatic perturbations. We discuss how the choice of frame in which perturbations are described can significantly simplify the mathematical analysis of the problem and show that it is possible to change frames directly from the linear level equations. We find explicitly that the case of isotropic, adiabatic perturbations can be reduced to a singular Sturm–Liouville eigenvalue problem, and lower bounds for the values of the eigenfrequencies can be derived. These results lay the theoretical groundwork to analytically describe linear, isotropic, and adiabatic perturbations of static, spherically symmetric spacetimes.

中文翻译：

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规范静态空间紧缩 LRS II 时空的不变扰动


我们提出了一个框架，用于描述广义相对论中静态、空间紧凑和局部旋转对称的 II 类时空的完全一般一阶扰动。扰动变量通过构造协变和识别规范不变，包括流体源的几何形状和热力学。然后将新方程应用于各向同性绝热扰动的研究。我们讨论了选择描述扰动的框架如何显着简化问题的数学分析，并表明可以直接从线性水平方程中更改框架。我们明确发现，各向同性绝热扰动的情况可以简化为奇异的 Sturm-Liouville 特征值问题，并且可以推导出特征频率值的下限。这些结果为分析描述静态、球对称时空的线性、各向同性和绝热扰动奠定了理论基础。